I spent a fair amount of time working it out.
It required constructing a parallelogram that incorporated the original triangle, parallel to the base and one side...
I didn't do it the way you illustrated.

I'll have to recheck my work (pretty scribbled at this point) and see what I did wrong.

Interesting approach but it seems like the parallelogram didn't buy you much information. Wondering if you created the parallelogram at point D instead, you could have found the angles needed to solve for x? Valiant effort" Worked on this for 3 days. I wasn't sure if I was correct until I added up the angles as a check.